First, let's make it clear exactly what variation of the game we are playing.
1. Prize is assigned to a random door with each door being equally likely. Neither you nor Monty know which door.
2. You pick a door. Because the prize was assigned randomly and you don't know where it is, it is irrelevant how you pick your door. Without loss of generality (WLOG) we can assume you always pick door #1.
3. The host picks a door and opens it. Because the prize was assigned randomly and Monty does not know where it is, it is irrelevant how Monty picks a door. WLOG we can assume he always opens door #2.
4. If Monty revealed the prize when he opened his door, the game ends and you lose.
5. If Monty did not reveal the prize, you are given the opportunity to switch to the remaining door (door #3).
6. Your door is opened. You win if the prize is behind it. Otherwise you lose.
There are three equally likely cases to consider.
1. The prize is behind door #1. This occurs 1/3 of the time. Monty opens #2. You are given the opportunity to switch. In this case switching is bad.
2. The prize is behind door #2. This occurs 1/3 of the time. Monty opens #2. The prize is there and the game ends. Note that in this case, YOU ARE NOT GIVEN THE OPPORTUNITY TO SWITCH.
3. The prize is behind door #3. This occurs 1/3 of the time. Monty opens #2. You are given the opportunity to switch. In this case switching is good.
Note that in the cases where you are given the opportunity to switch (#1 and #3), switching wins in one of them and switching loses in the other. Each of these cases is equally likely (occurring in 1/3 of the games of played), and so in this version of the game switching makes no difference.
Here's another way to look at it. Since neither you nor Monty know where the prize is when you pick doors, we could change the game so that the prize is not placed until AFTER Monty opens a door, and this would not change any probabilities.
So, in this modified but equivalent game, we play like this:
1. You pick a door.
2. Monty picks a door and opens it. There is nothing behind it, because the prize has not yet been placed.
3. You are asked if you want to switch to the other unopened door.
4. The prize is placed randomly.
5. If the prize is placed behind the opened door, the game ends and you lose.
6. Otherwise, your door is opened and you win if the prize is behind it.
It should be clear that you have a 1/3 chance of winning the car in this game no matter how you pick your door or whether or not you switch. At the time the prize is placed, there is a door that is now your door, and you win if and only if the prize gets randomly placed behind that door.
First, let's make it clear exactly what variation of the game we are playing.
1. Prize is assigned to a random door with each door being equally likely. Neither you nor Monty know which door.
2. You pick a door. Because the prize was assigned randomly and you don't know where it is, it is irrelevant how you pick your door. Without loss of generality (WLOG) we can assume you always pick door #1.
3. The host picks a door and opens it. Because the prize was assigned randomly and Monty does not know where it is, it is irrelevant how Monty picks a door. WLOG we can assume he always opens door #2.
4. If Monty revealed the prize when he opened his door, the game ends and you lose.
5. If Monty did not reveal the prize, you are given the opportunity to switch to the remaining door (door #3).
6. Your door is opened. You win if the prize is behind it. Otherwise you lose.
There are three equally likely cases to consider.
1. The prize is behind door #1. This occurs 1/3 of the time. Monty opens #2. You are given the opportunity to switch. In this case switching is bad.
2. The prize is behind door #2. This occurs 1/3 of the time. Monty opens #2. The prize is there and the game ends. Note that in this case, YOU ARE NOT GIVEN THE OPPORTUNITY TO SWITCH.
3. The prize is behind door #3. This occurs 1/3 of the time. Monty opens #2. You are given the opportunity to switch. In this case switching is good.
Note that in the cases where you are given the opportunity to switch (#1 and #3), switching wins in one of them and switching loses in the other. Each of these cases is equally likely (occurring in 1/3 of the games of played), and so in this version of the game switching makes no difference.
Here's another way to look at it. Since neither you nor Monty know where the prize is when you pick doors, we could change the game so that the prize is not placed until AFTER Monty opens a door, and this would not change any probabilities.
So, in this modified but equivalent game, we play like this:
1. You pick a door.
2. Monty picks a door and opens it. There is nothing behind it, because the prize has not yet been placed.
3. You are asked if you want to switch to the other unopened door.
4. The prize is placed randomly.
5. If the prize is placed behind the opened door, the game ends and you lose.
6. Otherwise, your door is opened and you win if the prize is behind it.
It should be clear that you have a 1/3 chance of winning the car in this game no matter how you pick your door or whether or not you switch. At the time the prize is placed, there is a door that is now your door, and you win if and only if the prize gets randomly placed behind that door.